Source Detection I:Theory

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Source Detection ITheory

• P. E. Freeman V. Kashyap
• 3rd X-ray Astronomy School
• 14 May 2003

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The Challenge Source Detection

• What challenge??

(Granted, it does look easy here. This is a
Hubble image of the globular cluster 47 Tuc,
courtesy P. Edmonds.)
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The Challenge Source Detection
It gets harder not everything seen in this
false-color Chandra image of the core for 47 Tuc
is an X-ray source. (P. Edmonds)
As seen in this ROSAT image of the Pleiades, the
X-ray source detector must worry about
high backgrounds and exposure variations (the
edge of the circular field-of-view and the
support-rib shadows). (V. Kashyap)
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The Challenge Source Detection

• Sources may be easy to see in a typical Hubble
  image. However, detecting and characterizing
  them becomes increasingly difficult at higher
  energies.
• Source data may consist of only a few counts,
  hence we must rely on the Poisson distribution
  when making statistical inferences.
• Some spatially extended sources (e.g., supernova
  remnants) emit brightly at high energies and may
  overlap with point sources, making detection and
  characterization of the latter more difficult.
• How a high-energy telescope blurs a point source
  (i.e., the telescopes point-spread function) may
  be spatially non-uniform.
• Is this a source, or a background fluctuation?

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Detection Theory the Short Form
Note the following statements are generic they
may or may not apply to the specific source
detection algorithm of your choice. They are
meant to build your intuition.

• The Ingredient(s) a N-dimensional event list or
  binned image. (And an exposure map, knowledge
  of the PSF, etc., if appropriate.)
• The Detection Tool some function that is
  localized (i.e., non-zero only over some
  characteristic scale) within at least some subset
  of the dimensions.
• The Hypotheses
• M0 the data in a given pixel are
  (Poisson-)sampled from the background.
• M1 the data are a sum of samples from the
  background and an astronomical source.

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The Five-Fold Path
For a given source detection algorithm, an
analyst might follow this five-fold path to
source detection Nirvana

• Select an appropriate function scale, s. (If one
  is attempting to detect a point source, this
  would be some encircled-energy radius of the
  PSF.)
• Estimate the background amplitude, B. (In
  actuality, one would do this estimation for each
  image pixel-here, we narrow the problem to a
  single pixel.)
• Determine the value of a selected model
  comparison test statistic, To.
• Determine the significance, a
• Compare a to a pre-determined threshold
  significance value ao.

If a lt ao, the pixel is associated with a source!
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Classic Detection CELLDETECT

• The Function(s) two box functions with unit
  amplitude, co-aligned and centroided at pixel
  (i,j). The number of counts within each box are
  Dd and Db.
• The Determination of B done by assuming (a) the
  truth of the alternative model, and (b) that the
  source is point-like
• where a and ß are the integrals of the PSF
  within each box, respectively.
• The Model Comparison Test Statistic the
  signal-to-noise ratio, or SNR
• Associating a Pixel with a Source If SNR gt
  SNRthr, accept the alternative model.
• For more information CIAO detect manual.

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New Detection WAVDETECT

• The Function the Marr, or Mexican Hat wavelet,
  W(s) (above, right), which is non-zero within a
  circle of radius ? 5s from the centroid.
• The Determination of B done by determining the
  average number of counts per pixel in the wavelet
  negative annulus (below, right), while using it
  as a weighting function done iteratively, with
  source counts removed from the field until the
  background estimate stabilizes.
• The Model Comparison Test Statistic
• To Co Si Sj W i-i,j-j D ij
• Associating a Pixel with a Source if
• a ?Co? dC p(C2ps2B) lt ao
• accept. A typical choice for the threshold
  is 1/P, where P is the number of pixels examined
  in the image it thus corresponds to a number of
  false pixels.

See Freeman et al. 2002, ApJS 138, 185 for more
details.
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Why Mexican-Hat Wavelets?

• The Gaussian-like positive kernel has a shape
  similar to canonical point-spread functions
  (PSFs).
• The function is localized in both the spatial and
  Fourier domains a dyadic (factors of two)
  sequence of scales is sufficient to sample the
  frequency domain.
• It has two vanishing moments the correlation
  of MH with constant and linear functions is zero.
  It thus annihilates the contribution of a
  spatially constant or linear background to the
  correlation coefficients.

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Potholes on the Five-Fold Path

• PSFs in the X-ray regime are spatially varying
  (which partially motivates the multi-scale
  approach to source detection the other major
  motivation is the study of extended sources,
  e.g., SNRs, hot gas in clusters).
• The optimal determination of B from raw data
  consisting of source and background counts is an
  unsurmounted statistical challenge.
• The cosmic background is not necessarily
  spatially constant!
• The probability sampling distributions (PSDs)
  from which observed values of T are sampled
  generally cannot be represented analytically,
  except asymptotically in the high-counts limit
  simulations are needed.
• There is no model comparison test statistic T
  that has been proven to be most powerfuland
  test power is extremely difficult to compute.
• And exposure maps, vignetting, etc. Vinay speaks
  of these.
• Below, I expand on some of these issues

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Pothole Spatially Varying PSFs

• A point source observed on-axis (center) with an
  X-ray telescope will be more sharply in focus
  than a source observed off-axis (outer eight
  panels), in large part because the
  counts-recording instruments are flat. Sources
  detected using a cell or wavelet of one scale may
  not be detected at another scale a multi-scale
  approach is necessary for robust detection!

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Pothole Background Determination
Strong source biases background in ring around it

• In theory, can model cosmicparticle background,
  but not easily done.
• Estimated from raw data. How to do?
• If one uses PSF information, one must make
  assumptions of spectral form (since width varies
  as function of energy) also, bad for detecting
  extended sources.
• WAVDETECT computes backgrounds at each scale, and
  combines them accurate enough for source
  detection, but systematic rings (top right) and
  bumps (bottom right) make final result not
  necessarily quanitatively accurate.

Large-scale source biases background at source
location
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Pothole Spatially Varying Backgrounds

• Nearby regions of dense gas/dust can absorb the
  cosmic background (e.g., from AGNs), creating
  X-ray shadows such as that observed in the
  Pleiades. (Nearby emission in the hot local
  bubble means that we still see background photons
  in shadowed regions.)

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Pothole Computation of Significance

• In the high-count limit, p(Cq 2pB2) tends to a
  zero-mean Gaussian distribution of width s
  q1/2.
• Elsewhere, p(Cq) is determined via simulations.
  (At right, a sample PSD at low counts from
  Damiani et al. 1997.)

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Pothole Computation of Significance

• The picture to the right shows the significance
  (from simulations) as a function of q and C (to
  the left of the cusp) or C/sqrt(q) (to the right
  of the cusp). The contours are 0.1 (bottom) to
  10-7 (top). This figure demonstrates that a
  relatively smooth distribution (asymptotically
  Gaussian, at high q) becomes very messy at low q,
  and shows that simulations are required. The low
  q limit is important for Chandra, whose the
  smaller field of view greatly reduces the number
  of cosmic background events per pixel per second,
  relative to ROSAT.

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Pothole Type II Error

• The Type II error is nearly impossible to compute
  for current source detection algorithms because
  of the fuzzy way the problem is stated the
  alternate hypothesis is that pixel (i,j)
  includes some number of source counts.
• Computed instead is the detection efficiency
  how often does the algorithm detection a source
  of strength x, at off-axis location y, when the
  background is z

• Unlike Type I error, detection efficiency is
  instrument-specific.
• Depends on scale sizes, background, amplitude,
  extent, spectrum, and off-axis angle, in addition
  to the details of the exposure at the source
  location.

• A related topic upper limits (I dont detect a
  source at (i,j). How strong could an underlying
  source be and still not be detected?). Rarely
  analytically computable, it can be read off from
  detection efficiencies, if those have been
  computed.